# Full text of "Analytical Mechanics"

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```UNIPLANAR MOTION OF A RIGID BODY £ =     Ma2 C Ma
231
. # sin a;.
Thus both the linear acceleration and the angular acceleration are constant. Therefore the equations of motion may be obtained as in the preceding problem.
DISCUSSION.
ma
.
2^2
•-M-
a4-
2 (a2 + 62) Substituting this value of Ic in the expression for v we get
Case 7. — Let 6 = a, then v = f # sin a, which is the acceleration of a cylinder rolling down an inclined plane.
2 Case II. — Let 6 <C a, then ^= - g sin a, as in case I.
Caselll. — Let b » a, then y =
2 a2
sin a.   Thus by reducing the
radius of the axle we can reduce the acceleration, theoretically at least, as
much as we please.   The reason for this fact becomes clear when we con-
sider the relative proportions of
the potential energy which are
transformed into kinetic energy
of translation and kinetic energy
of rotation.
3. In Fig. 117 the larger circle represents a cylinder of mass M which rolls along a rough horizontal table, under the action of a falling body of mass m. The right-hand end of the ribbon, which connects the falling body with the cylinder, is wound around the latter so that it is unwound as the motion goes                          _ __
rrn          n                  i •   i    xl_                                   FlG-   H7.
on.   The pulley over which the
ribbon slides is smooth.   Discuss the motion, supposing the mass and
the thickness of the ribbon to be negligible.
```